A decision tree as in Fig. 1b in Kampichler & Platen (2004) is
a classifier with discrete decision criteria. For example, a moor with
50 individuals of species A and with 101 individuals of species B in a
sample of a defined size would be classified as belonging to class X. If
only one specimen of species B was missed, the moor would be assigned to
another class, Y, despite the obvious similarity between the two cases.
Where ecologists have previously drawn this kind of artificially sharp
distinction, they can now draw more realistic boundaries by means of fuzzy
sets. In classical ('crisp') set theory, there are only two possibilities;
either an object is member of a set or is not; thus, the only possible membership
values are 0 and 1. In the example above, the moor has the membership 1
in class X (is a member of the class) and the membership 0 in class Y (is
not a member of the class).The central idea in fuzzy set theory is that
members of a set may have only partial membership, which consequently may
possess all possible values between 0 and 1. The closer the membership of
an element is to 1, the more it belongs to the set; the closer the membership
of an element is to 0, the less it belongs to the set. Let, for example,
the possible numbers of individuals of a species in a defined sample lie
within 0 and 100 ( Fig. A
). Sharp boundaries between sets necessarily mean, that counts that
differ as close as 1 may be assigned to different sets (here, 50 and 51).
Through fuzzy sets, a region of overlap may be defined; numbers around 50
belong to both sets, the respective membership values depending on whether
the observed numbers are lower or larger than 50.

The decision trees yielded by automated tree induction were broken down
into a set of rules by representing each possible path through the trees
by a rule. (See5 uses a more parsimonious approach for generating rules out
of a tree; however, for a fuzzy model, the rule base must explicitly cover
the entire variable space.) Subsequently, sharp boundaries (e.g., a rule
including the antecedent IF species A <= n versus a rule including IF
species A > n, where n defines a split in the tree) were translated into
fuzzy boundaries (e.g. into rules with the antecedents IF species A is rare
versus IF species A is frequent with an overlapping zone between the sets
"rare" and "frequent" around n) in a manner similar to that demonstrated in
Fig. A. The amount of overlap chosen was based on biological plausibility
and was open to modification while adjusting the model. Fuzzy models can be
tuned by modifying the shape of the fuzzy sets until the model output eventually
fits the desired output. Presence-absence splits in the decision trees were
translated into discrete sets. For development of the fuzzy model, we chose
only 15 of the plots according to a stratified random drawing from the full
list of plots in order to ensure an equal representation of each degradation
stage in the subset; we used the ten plots not chosen for development as
unseen cases for validation of the model (Rykiel 1996).

For a hypothetical example, let there be two species (A and B) and
two fuzzy sets denoting the abundance of A and B ("rare" and "frequent")
( Fig. B
), and let X, Y and Z be three classes of increasing moor degradation.
There are four possible combinations of variable states, and they can be
described by four rules (as it would be if the sets were discrete); let them,
hypothetically, be:

Rule 1: IF species A is rare and species B is rare THEN moor belongs
to class X.
Rule 2: IF species A is rare and species B is frequent THEN moor belongs
to class Y.
Rule 3: IF species A is frequent and species B is rare THEN moor belongs
to class Z.
Rule 4: IF species A is frequent and species B is frequent THEN moor
belongs to class Z.

Logically, rules 3 and 4 could be replaced by the simpler rule IF species
A is frequent THEN moor belongs to class Z, but the procedures involved
in running a fuzzy rule-based model need the explicit addressing of all
possible variable states. Let the observed numbers n of the two species
be n A=10 and nB
=60. In a process called fuzzification (Fig. B
), the observed numbers of individuals of A and B are translated into
membership values in the sets "rare" and "frequent". Of the four rules,
rule 3 and 4 are not activated, since nA
=10 is not a member of the class "frequent", the respective membership
value is 0. Both rule 1 and 2 are activated, because -- due to the fuzzy nature
of the sets -- n B=60 belongs to set "rare"
as well as to set "frequent" and this is a fundamental difference to a rule
base with discrete sets: Whereas in a discrete rule-set each possible combination
of attributes is represented by only one rule, in a fuzzy rule-set several
rules may address the same combination of attributes.

Fig. B. Hypothetical example for fuzzy
control consisting of the processes fuzzification, fuzzy inference and
defuzzification. The abundance of species A and B is 10 and 60 individuals,
respectively. See text for detailed description.

In a process called fuzzy inference (Fig. B
), the membership values of the moor in the classes X, Y and Z are
calculated. This is done by use of the minimum-operator: the lower value
of the membership values of A and B is assigned to the class that is addressed
by the rule (this is the simplest possibility for the logical AND in fuzzy
logic). Subsequently, the results of rule 1 and 2 are merged by use of the
maximum-operator: the higher value of the membership determined by the
different rules is assigned to each class (this is the simplest possibility
for the logical OR in fuzzy logic) (Fig. B
). In a final step called defuzzification (Fig.B
), the result of the fuzzy inference can be transformed (when necessary)
into a discrete output. The method most widely used is to calculate the
centre of gravity of the output polygon and to project it onto the x-axis.
This value can be used for assigning a case (i.e., moor) to a certain class
(e.g., the x co-ordinate of the centre of gravity is closest to class Y,
this means that the moor with nA=10 and
n B =60 is assigned to class Y). Thus,
classes must be expressed at least on an ordinal scale.

The entire process embracing fuzzification, fuzzy interference and
defuzzification is called fuzzy control. The simple example above shows
only a few of the possibilities available for fuzzy control; the operators
used and the shape of the fuzzy sets (trapezoid, triangular, overlapping
vs. non-overlapping etc.) are subject to the expertise of the modeller
(see Bothe (1995) or Zimmermann (1996) for an introduction to fuzzy control).

a stenotopic, here: occurring in not more than two habitat
types (e.g., oligotrophic and mesotrophic mires)b eurytopic, here: occurring in more than seven habitat types
of any kindc intermediate, between stenotopic and eurytopicd Many Bembidion species are known as insect egg predators
.

The table "Fuzzy sets" shows the intersection points of the set boarders
with the isolines "membership = 0" and "membership = 1". For example,
the triangle representing the fuzzy set "is_absent" for Agonum afrum
is is defined by the lower left point (0/0), the upper point (0/1) and
lower right point (1/0). In analogy, trapezoid sets are defined by four
points. The numbers in parentheses behind the names of the fuzzy sets
are used for listing the rules in the table "Rule base".

Table B.1. Fuzzy sets

Species

Fuzzy Set

Shape

Points

Agonum afrum

is_absent (1)

triangle

[0 0 1]

is_present (2)

trapezoid

[0 1 500 500]

Amara lunicollis

is_rare_or_absent (1)

triangle

[0 0 4]

is_present (2)

trapezoid

[0 4 500 500]

Pterostichus diligens

is_moderately_abundant_or_absent (1)

trapezoid

[0 0 10 50]

is_abundant (2)

trapezoid

[10 50 110 176]

is_very_abundant (3)

trapezoid

[110 176 1000 1000]

Calathus micropterus

is_absent (1)

triangle

[0 0 1]

is_present (2)

trapezoid

[0 1 500 500]

Table B.2. Rule-base

Rule no.

A. afrum

A. lunicollis

P. diligens

C. micropterus

Degradation stage

1

2

1

1

1

1

2

2

1

1

2

1

3

2

1

2

1

2

4

2

1

2

2

4

5

2

1

3

1

3

6

2

1

3

2

3

7

2

2

1

1

5

8

2

2

1

2

5

9

2

2

2

1

5

10

2

2

2

2

5

11

2

2

3

1

5

12

2

2

3

2

5

13

1

1

1

1

5

14

1

1

2

1

5

15

1

1

2

2

5

16

1

1

2

2

5

17

1

1

3

1

5

18

1

1

3

2

5

19

1

2

1

1

5

20

1

2

1

2

5

21

1

2

2

1

5

22

1

2

2

2

5

23

1

2

3

1

5

24

1

2

3

2

5

For example, rule 1 is to be read as

IF Agonum afrum is_present AND IF Amara lunicollis
is_rare_or_absent AND IF Pterostichus diligens is_moderately_abundant_or_absent
AND IF Calathus micropterus is_absent THEN moor belongs to degradation
stage 1.